Conserved quantities in the Doran Metric?

In summary: Sam Gralla notes, by the symmetries of the spacetime. These can be read off, roughly speaking, by looking at the metric and seeing which coordinates it doesn't depend on. (I say "roughly speaking" because you need to be using the right set of coordinates for this to work. Strictly speaking, the symmetries are captured by the Killing vectors of the spacetime; the "right" set of coordinates is one where as many coordinates as possible match up with Killing vectors.)Some remarks:If a metric has explicit time symmetry, i.e. t does not appear in the metric, then the vector with components <1,0
  • #1
m4r35n357
658
148
I've been doing some amateur simulations of particle trajectories in a few well-known metrics (using GNU Octave and Maxima), and in the case of the Schwarzschild and Gullstrand-Painleve metrics I have the ability to check my results using freely available equations for conservation of energy and angular momentum.

In the case of the Doran metric however, the geodesic equations are far messier(!), but I believe I now have a "correct" simulation. I would like to check this also, but have not seen any reference to equations for conserved quantities for this metric.

Has anyone here looked into this, or does anyone know any useful links, or are there any computer algebra wizards that can help?
 
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  • #2
What's the Doran metric? If it doesn't have (m)any symmetries, don't hold your breath for conserved quantities. If it does, work them out yourself: the dot product of a killing vector with the four-velocity is conserved along a geodesic.
 
  • #3
By the "Doran metric", do you mean the Doran chart on Kerr spacetime? This chart is described, for example, by Matt Visser in this article (Section 7):

http://arxiv.org/abs/0706.0622

If that is what you mean, then the conserved quantities are dictated, as Sam Gralla notes, by the symmetries of the spacetime. These can be read off, roughly speaking, by looking at the metric and seeing which coordinates it doesn't depend on. (I say "roughly speaking" because you need to be using the right set of coordinates for this to work. Strictly speaking, the symmetries are captured by the Killing vectors of the spacetime; the "right" set of coordinates is one where as many coordinates as possible match up with Killing vectors.)
 
  • #4
Some remarks:

If a metric has explicit time symmetry, i.e. t does not appear in the metric, then the vector with components <1,0,0,0>, which we will call [itex]\xi^a[/itex] is a Killing vector of that metric.

And we can write the dot product [itex]E = g_{ab} \, \xi^a \, u^b[/itex] = constant along any geodesic, [itex]u^b[/itex] being the tangent vector of that geodesic at any point, the same [itex]u^b[/itex] that appears in the geodesic equation.

To confirm this (including to prevent any errors by confusing vectors with one forms or other mistakes in interpreting the notation, or typos in my post) you should be able to confirm that one of the geodesic equations is equivalent to [itex] d E/ d \tau = 0[/itex], where you expand the total derivative (dE/d \tau), which is taken to be along the geodesic curve which is parameterized by [itex]\tau[/itex] as usual using the chain rule.

Example: for the Schwarzschild metric we can write , for some geodesic parameterized by [itex]\tau[/itex] so that we have the geodesic curve [itex]t(\tau), r(\tau), \theta(\tau), \phi(\tau)[/itex], and the tangent vector [itex]u^a[/itex] of this curve with components [itex] u^t = dt/d\tau, u^{r} = dr/d \tau, u^{\theta} = d \theta / d \tau, u^{\phi} = d \phi / d \tau [/itex]

[tex]
E(\tau) = -(1-2m/r) u^t = -\left(1 - \frac{2m} {r(\tau)} \right) \left( \frac{ d \, t(\tau) } { d \, \tau } \right)
[/tex]

and we can expand using the chain rule

[tex]
dE / d\tau = -(1-2m/r) ) \frac{d^2 t}{d \tau^2}- \frac{2m}{r^2} \left( \frac{d \, r } {d \tau} \right) \left( \frac{d \, t} {d \tau} \right) = 0
[/tex]

Which with some algebra this can be seen to be identical to one of the standard geodesic equations for the Schwarzschild metric.

Thus we see that one of the geodesic equations is equivalent to a statement that E is constant along a geodesic curve.

Furthermore, if you convert to a coordinate chart that doesn't have explicit time symmetry, the tensor equations above will still be true, but the Killing vector will have different components. And because the Killing vector transforms like a vector, you can use the standard vector transformation laws to find the components in the new coordinate chart - so you can convert the Kerr Killing vectors to the Doran chart, for instance.
 
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  • #5
OK thanks for the replies so far, for information my main link for the metric is arxiv.org/pdf/gr-qc/0411060 (sorry but the whole link is too messy). You might be able to imagine that the Christoffel symbols (34 non-unique) are hideous to work with (at least the output from Maxima, which is very poorly factorized) so I don't fancy my chances at much of the maths on my own, hence my question here. I don't have GRTensor and I just hoped that it would do a better job . . . .
Pervect: I will have to take my time to absorb your comments fully, I think you have given me some good information, thank you.
 
  • #6
m4r35n357 said:
OK thanks for the replies so far, for information my main link for the metric is arxiv.org/pdf/gr-qc/0411060

This is the same Doran metric that is treated in the paper I linked to; the two treatments give pretty much the same information.
 
  • #7
m4r35n357 said:
I've been doing some amateur simulations of particle trajectories in a few well-known metrics (using GNU Octave and Maxima), and in the case of the Schwarzschild and Gullstrand-Painleve metrics I have the ability to check my results using freely available equations for conservation of energy and angular momentum.

In the case of the Doran metric however, the geodesic equations are far messier(!), but I believe I now have a "correct" simulation. I would like to check this also, but have not seen any reference to equations for conserved quantities for this metric.

Has anyone here looked into this, or does anyone know any useful links, or are there any computer algebra wizards that can help?

You might find the following links useful-

'A new form of the Kerr solution' Chris Doran
http://lanl.arxiv.org/abs/gr-qc/9910099

'Painleve-Gullstrand Coordinates for the Kerr Solution' Jose Natario
http://arxiv.org/abs/0805.0206v2
 
  • #8
stevebd1 said:
You might find the following links useful-

'A new form of the Kerr solution' Chris Doran
http://lanl.arxiv.org/abs/gr-qc/9910099

'Painleve-Gullstrand Coordinates for the Kerr Solution' Jose Natario
http://arxiv.org/abs/0805.0206v2

Cheers for those, think I need to take a break from doing and go back to reading for a short while . . .
 
  • #9
pervect said:
and we can expand using the chain rule
erm, sorry to sound stupid, but I'm sort of seeing this example as the product rule, do you really mean the chain rule?
[EDIT] OK I see both now, np ;)
 
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  • #10
pervect said:
Furthermore, if you convert to a coordinate chart that doesn't have explicit time symmetry, the tensor equations above will still be true, but the Killing vector will have different components. And because the Killing vector transforms like a vector, you can use the standard vector transformation laws to find the components in the new coordinate chart - so you can convert the Kerr Killing vectors to the Doran chart, for instance.

Firstly, thanks for the clearest description I have heard regarding Killing vectors and conserved quantities in GR. With your help I now have two conserved quantities in my Doran metric simulations. It turned out that my initial guess for the expressions was right, I just had an error in the implementation, duh. But from your comments I now know why my guess was the right thing to do . . .

I am also now intrigued at the possibility of replacing two huge geodesic equations with much simpler expressions, Maxima permitting ;)

Still working my way through the last part of your post (quoted). As it happens I need to implement some control over coordinates in order to set initial conditions in a consistent way.
 

Related to Conserved quantities in the Doran Metric?

1. What are conserved quantities in the Doran Metric?

Conserved quantities in the Doran Metric refer to physical quantities that remain constant throughout the evolution of a system described by the Doran Metric. These quantities are independent of the choice of coordinates and are associated with symmetries of the Metric.

2. What are some examples of conserved quantities in the Doran Metric?

Examples of conserved quantities in the Doran Metric include energy, momentum, and angular momentum. These quantities are conserved due to the symmetries of the Metric which describe the behavior of physical systems.

3. How are conserved quantities related to symmetries in the Doran Metric?

Conserved quantities in the Doran Metric are associated with symmetries of the Metric. This means that when the Metric is invariant under certain transformations, there exists a conserved quantity that remains constant throughout the evolution of the system. This is known as Noether's theorem.

4. Can conserved quantities in the Doran Metric change?

No, conserved quantities in the Doran Metric remain constant throughout the evolution of a system. This is because the Metric is invariant under certain transformations, meaning that these quantities are preserved and do not change over time.

5. How do conserved quantities in the Doran Metric affect the behavior of physical systems?

Conserved quantities in the Doran Metric play a crucial role in determining the behavior of physical systems. These quantities provide a measure of the system's symmetry and can be used to predict its dynamics and evolution. They also help to understand the fundamental laws of physics, such as the conservation of energy and momentum.

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